Editing Inverse element (section)

=== ''U''-semigroups ===

A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (''a''°)° = ''a'' for all ''a'' in ''S''; this endows ''S'' with a type {{langle}}2,1{{rangle}} algebra. A semigroup endowed with such an operation is called a '''''U''-semigroup'''. Although it may seem that ''a''° will be the inverse of ''a'', this is not necessarily the case. In order to obtain interesting notion(s), the unary operation must somehow interact with the semigroup operation. Two classes of ''U''-semigroups have been studied:<ref>Howie p. 102</ref>

* '''''I''-semigroups''', in which the interaction axiom is ''aa''°''a'' = ''a''
* '''[[Semigroup with involution|*-semigroups]]''', in which the interaction axiom is (''ab'')° = ''b''°''a''°. Such an operation is called an [[involution (mathematics)|involution]], and typically denoted by ''a''*

Clearly a group is both an ''I''-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are [[completely regular semigroup]]s; these are ''I''-semigroups in which one additionally has ''aa''° = ''a''°''a''; in other words every element has commuting pseudoinverse ''a''°. There are few concrete examples of such semigroups however; most are [[completely simple semigroup]]s. In contrast, a subclass of *-semigroups, the [[Semigroup with involution#Drazin|*-regular semigroup]]s (in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the [[Moore–Penrose inverse]]. In this case however the involution ''a''* is not the pseudoinverse. Rather, the pseudoinverse of ''x'' is the unique element ''y'' such that ''xyx'' = ''x'', ''yxy'' = ''y'',   (''xy'')* = ''xy'', (''yx'')* = ''yx''. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the ''generalized inverse'' or ''Moore–Penrose inverse''.